
where \(\widehat{\mathsf {iA}}\)  is an appropriate preconditioner of the SPD matrix
\(\mathsf {A}:=\mathsf {A}_{h}^\mu +\frac{2}{\delta t}\mathsf {M}^{\rho }_h,\) 
and
\(\widehat{\mathsf {iS}}\)  is an appropriate preconditioner of the
(dense) Schur complement SPD matrix
\(\mathsf {S} :=\mathsf {B}_h^T(\mathsf {A}_{h}^\mu +\frac{2}{\delta t}\mathsf {M}^{\rho }_h)^{-1}\mathsf {B}_h+\mathsf {M}_h^{\gamma }.\) 
The detailed construction of the
preconditioner for the Schur complement (pressure) matrix \(\mathsf {S}\)  is discussed in Subsection REF , where
we borrow ideas in the literature on preconditioning
the closely related, generalized Stokes problem [1]}, [2]}, [3]}.
The detailed construction of the
preconditioner for the SPD velocity matrix \(\mathsf {A}\) 
is discussed in Subsection REF , where
we use
an auxiliary space preconditioner [4]}
along with algebraic multigrid.
We mention that for polynomial degree \(k\ge 2\) , the preconditioned
MinRes solver is applied to the static condensed subsystem of
(REF ), see the discussion in Remark REF .
